A point P lies on the circle x2 + y2 = 169. If Q = (5, 12) and R

Subject

Mathematics

Class

JEE Class 12

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 Multiple Choice QuestionsMultiple Choice Questions

1.

Each of a and b can take values 1 or 2 with equal probability. The probability that the equation ax2 + bx + 1= 0 has real roots, is equal to

  • 12

  • 14

  • 18

  • 116


2.

Cards are drawn one-by-one without replacement from a well shuffled pack of 52 cards. Then, the probability that a face card (jack, queen or king) will appear for the first time on the third turn is equal to

  • 3002197

  • 3685

  • 1285

  • 451


3.

There are two coins, one unbiased with probaility 12 or getting heads and the other one is biased with probability 34 of getting heads. A coin is selected at random and tossed. It shows heads up. Then, the probability that the unbiased coin was selected is

  • 23

  • 35

  • 12

  • 25


4.

Lines x + y = 1 and 3y = x + 3 intersect the ellipse x2 + 9y2 = 9 at the points P,Q and R. The area of the PQR is

  • 365

  • 185

  • 95

  • 15


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5.

For the variable , the locus of the point of intersection of the lines 3tx - 2y + 6t = 0 and 3x + 2ty - 6 = 0 is

  • the ellipse x24 + y29 = 1

  • the ellipse x29 + y24 = 1

  • the hyperbola x24 - y29 = 1

  • the hyperbola x29 - y24 = 1


6.

The locus of the mid-points of the chords of an ellipse x2 + 4y2 = 4 that are drawn from the positive end of the minor axis, is

  • a circle with centre 12, 0 and radius 1

  • a parabola with focus 12, 0 and directrix x = - 1

  • an ellipse with centre 0, 12, major axis 12 and minor axis

  • a hyperbola with centre 0, 12, transverse axis 1 and conjugate axis 12


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7.

A point P lies on the circle x2 + y2 = 169. If Q = (5, 12) and R = (-12, 5) then the QPR is

  • π6

  • π4

  • π3

  • π2


B.

π4

Given equation of circle is,

x2 + y2 = 169

Its centre = ( 0, 0) and radius = 13

Now, slope of OR = - 512 = m1            slope = y2 - yx2 - x

 and slope of OQ = 125 = m2

 m1 . m2 = - 1 ROQ = π2

We know that, angle made by the chord of circle at circumference is equal to the half of the angle made by the same chord at the centre of circle.

 QPR = 12 × π2 = π4


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8.

A point moves, so that the sum of squares of its distance from the points (1, 2) and (- 2, 1) is always 6. Then, its locus is

  • the straight line y - 32 = - 3x + 12

  • a circle with centre - 12, 32 and radius 12

  • a parabola with focus (1, 2) and directrix passing through (- 2, 1)

  • an ellipse with foci (1, 2) and (- 2, 1)


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9.

A circle passing through (0, 0), (2, 6), (6, 2) cut the x-axis at the point P  (0, 0). Then, the lenght of OP, where O is the origin, is

  • 52

  • 52

  • 5

  • 10


10.

For the variable t, the locus of the points of intersection of lines x - 2y = t and x + 2y = 1t is

  • the straight line x = y

  • the circle with centre at the origin and radius 1

  • the ellipse with centre at the origin and one focus 25, 0

  • the hyperbola with centre at the origin and one 52, 0


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